Modelling Host-Parasite Coevolution in Continuous Space

Coevolution can lead to

spatial mosaics of phenotypic and genomic diversity

• Classically: interspecific spatial correlation \(\implies\) coevolution

  • Not true since other processes can generate correlation, limited dispersal in particular

Two projects

focused on disentangling spatial patterns due to

coevolution, limited dispersal and drift

• Phenotypic (math)

• Genomic (slim)

• Focusing on phenotypic model today

Goals

• Approximate spatial scales of phenotypic turnover and coadaptation
• Introduce measures of local adaptation for continuously distributed populations
• Use results to determine which species is locally adapted
  • As function of relative dispersal abilities and strengths of selection
• Test accuracy of math using simulations (not slim)
  • Benefit: allows us to relax key model assumptions
  • Cost: mapping simulation parameters to math parameters is hard
    • e.g. no such thing as interaction radius in math model

The model

• Tracks (co)evolution of mean traits across 2D habitat
• Drift captured by Gaussian noise in mean trait evolution
• Dispersal is Gaussian in both species
• Single trait mediates interaction/coevolution
  • Trait matching/mismatching

Key assumptions
(to be relaxed)

• Spatially homogeneous:
  • abundance densities
  • additive genetic and phenotypic variances
• Weak coevolutionary selection

What it looks like

Methods

• Apply Fourier theory to model to approximate:
  • Spatial (intraspecific) auto-covariance functions
    • Provides notion of scale for within species spatial patterns
  • Spatial (interspecific) cross-covariance function
    • Provides notion of scale for between species spatial patterns

Spatial scales of phenotypic turnover

• Spatial (intraspp) auto-cov functions are approximately Matérn

  • Matérn comes with built-in notion of spatial scale:

• \[\xi_S = \frac{\sigma_S}{\sqrt{G_S\psi_S}}\]

• \(\sigma_S=\) dispersal distance of species \(S=H,P\)

• \(G_S=\) additive genetic variance

• \(\psi_S=\) cumulative strength of selection

  • \(\psi_H=A_H-B_H, \ \ \ \psi_P=A_P+B_P\)

Spatial scale of coadaptation?

• Spatial (interspp) cross-cov function analytically intractable
• How to measure spatial scale of coadaptation?
• Plan:
  1. Numerically compute cross-cov function
  2. Fit a widely-employed spatial cov function to theoretical result
  3. Fit same function to simulated results to assess accuracy

Local adaptation

• Can quantify local adaptation using collocated covariance

• \(\ell_H = -B_H\mathrm{Cov}(\bar z_H,\bar z_P)\)

• \(\ell_P = B_P\mathrm{Cov}(\bar z_H,\bar z_P)\)

• \(B_H,B_P=\) strengths of biotic selection

• \(\bar z_H,\bar z_P=\) local mean traits

Local adaptation as function of relative dispersal distances

Testing theory with simulations

• Individual-based simulations
• Requires fitness functions (expd offspg num)
  • Theory uses growth rates
  • How to relate?
• Math! (theory of diffusion-limits in particular)

Mapping parameters

Additive genetic variance

\[G_P\approx \sqrt{\mu_P/(A_P+B_P)}\]

Population density

\[\rho_P\approx \frac{1}{\ln\kappa_P}\Big(r_P-\frac{1}{2}\sqrt{\mu_P(A_P+B_P)}\Big),\]

Growth rate

\[r_P\approx\ln\alpha_P+\pi_{\max}(\iota_P-1)\big(1-e^{-2\pi R_\iota^2\rho_H}\big),\]

Strength of biotic selection

\[B_P\approx\gamma\pi_{\max}(\iota_P-1)\big(1-e^{-2\pi R_\iota^2\rho_H}\big)\]

How well does it do?

• To evaluate, toss down 20 disks for each spp

  • Radius equal to competition radius for each spp
  • Measure population density and additive genetic var within each disk
  • Average across disks for an estimate

• Result:

  ##          Parameter Expectation Observation
  ## 1     host density   12.888329   13.650000
  ## 2 parasite density    8.707539    7.750000
  ## 3           host G    8.689459    3.892073
  ## 4       parasite G    6.932665    2.789218